Draining the vat. With only the small valve open, all of the liquid can be drained from a large vat in 4 hours. With only the large valve open, all of the liquid can be drained from the same vat in 2 hours. How long would it take to drain the vat with both valves open?

rate1=vat/2hrs

rate2=vat/4hrs

so in 4 hours, three vats could be emptied.

combined rate= 3 vats/4hrs

time= 1vat/combinedrate= 1.333 hrs

To find out how long it would take to drain the vat with both valves open, we need to determine the rate at which liquid is drained when both valves are open.

Let's consider the rates at which the liquid drains when only the small valve or only the large valve is open. If the entire vat can be drained through the small valve in 4 hours, it means that the small valve drains 1/4 of the vat's contents per hour. Similarly, if the entire vat can be drained through the large valve in 2 hours, it means that the large valve drains 1/2 of the vat's contents per hour.

Now, let's consider what happens when both valves are open. Since both valves are draining simultaneously, their individual rates of drainage should add up. Therefore, the combined rate of drainage when both valves are open is the sum of the rates when each valve is open.

So, the combined rate of drainage when both valves are open is 1/4 + 1/2 = 3/4 of the vat's contents per hour.

To find the time it would take to drain the entire vat with both valves open, we can divide the total contents of the vat by the combined rate of drainage:

Draining time = Vat contents / Combined rate of drainage = 1 / (3/4) = 4/3 hours.

Therefore, it would take 4/3 hours, or 1 hour and 20 minutes, to drain the vat with both valves open.